Factoring a quadratic equation may seem super easy to you now as you step into the world of cubic equations. Watch this video lesson to learn one easy method that you can use to factor some cubic equations.
Cubic equations are similar to your quadratic equations except that their highest degree is three instead of two. For example, 2x^3 + 4x^2 + x + 2 = 0 is an example of a cubic equation. Notice the little three that is your highest exponent.At first glance, this equation looks like it would be impossible to factor using any of the methods that you used for quadratic equations.
And you know what? It’s true.Cubic equations are in a whole other category, and it requires different methods to factor them. But don’t worry – it isn’t all that difficult. This video lesson will teach you one of the methods that you can use to easily factor some cubic equations. So keep watching!
The method that we will be learning to use is called the grouping method. It involves factoring groups of two terms to find a common factor.
If you view each term as a little person with a brain, then you can think of this method as combining two heads together to help you think even better. Aren’t two heads better than one? Won’t you have more brain power?Once you start this method, you will need to make use of the factoring skills you already know. You will need to know how to look for the greatest common factor and, then, to factor it out. You will also need to make use of your skills at factoring quadratics.
If you are a bit rusty on these, then take some time right now to review them. You can pause this video while you review. So now, let’s begin.
We will look at two examples to see how this grouping method helps you and why it is like combining two heads for increased brainpower.We will first tackle the cubic equation we saw in the beginning of this video: 2x^3 + 4x^2 + x + 2 = 0.The first step in the grouping method is to separate the terms into groups of two. Think of combining two heads. We will use parentheses to mark our groups. We will separate our groups with plus signs. So, remember to keep the sign of each term with that term.
(2x^3 + 4x^2) + (x + 2) = 0 is what we get after grouping.Now we’re going to factor each group by taking out the greatest common factor in each. The first group has a greatest common factor of 2x^2 that we can take out. The second group doesn’t have one that we can take out, so it stays the same.
We get 2x^2(x + 2) + (x + 2) = 0 after factoring our greatest common factor in each. Do you see something interesting here? That’s right! We have a common factor of (x + 2) in both groups. This tells us that we can rewrite this in the form of two parentheses multiplied together.One of our parentheses is the (x + 2) and the other is made from the greatest common factors we took out. Since our last group didn’t have a greatest common factor, it will be a one.
Remember to always keep the sign with each term. We don’t have negatives here, but if we did, we would be careful to keep each negative sign with the proper term.We rewrite it as (2x^2 + 1)(x + 2) = 0. We see that we now have a quadratic in one parentheses and a finished factor in the other. Now we need to ask ourselves if the quadratic can be factored more. If we use our quadratic factoring skills to analyze this, our answer is no; we can’t factor this anymore.
That means I am done factoring. My answer, then, is (2x^2 + 1)(x + 2) = 0.
Now let’s try another example. Factor x^3 + x^2 – x – 1 = 0.
See if you can work this one out on your own. Pause this video for a bit while you work it out and then continue to see if you’ve gotten it right.First, we group the terms by twos: (x^3 + x^2) + (-x – 1) = 0.
We use a plus to separate our groups and we make sure that we keep the proper sign with each term.Next, we factor out the greatest common factor in each group: x^2(x + 1) – 1(x + 1) = 0. We rewrite it as two sets of parentheses multiplied together: (x^2 – 1)(x + 1) = 0.Now we look at our quadratic to see if we can factor it further. The answer is yes. So we go ahead and do that.(x + 1)(x – 1)(x + 1) = 0.
We are now done since we can’t factor any of these further. We are done, but we could rewrite this in a different way if we wanted to.We see that we have (x + 1) twice, so we can combine them to become (x + 1)^2. So, we can rewrite our answer to be (x + 1)^2 * (x – 1) = 0. Either way will work.
Look for both when working with multiple-choice problems, as either can be shown.
Now, let’s review what we’ve learned. We learned that we can factor cubic equations, equations whose highest degree is three, using the grouping method, which involves factoring groups of two terms to find a common factor.If we think of each term as a little person with a head, then we can think of the grouping method as combining two heads together to help us factor more smartly.
We group first by forming our groups of two terms using parentheses. We separate each set of parentheses with a plus term. We make sure that we keep the proper sign with each term.We then factor out the greatest common factor in each group. If we did it correctly, then we will see that we have a common factor in both groups.
Next, we rewrite our greatest common factors in one set of parentheses and our common factor in another set of parentheses. These parentheses will be multiplied together.Next, we use what we know about factoring quadratics to see if we can continue factoring. Once we can’t factor anymore, then we are done.
Review this educational video lesson so that you can:
- Factor a cubic equation using the grouping method
- Separate terms and eliminate the greatest common factor in each group
- Identify and rewrite the greatest common factors